Subgroups — #29600191

Order 15616.

238 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16839595	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₂×D₆₁	C₂	1	1	yes	maximal, characteristic
3	#21447414	C₂	1	1	yes	maximal, characteristic
4	#21443953	C₂	1	1	yes	maximal, characteristic, complemented
5	#21443952	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₂×Dic₆₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14519716	C₄	1	1	yes	characteristic
9	C₂×C₁₆×D₆₁	C₂²	1	1	yes	characteristic
10	#16839716	C₄	1	1	yes	characteristic
11	#16839595	C₄	1	4	yes	complemented
12	C₂×#14519683	C₂²	1	1	yes	characteristic
13	C₂×#14519683	C₂²	1	1	yes	characteristic
14	C₃₂×D₆₁	C₄	1	4	yes	complemented
15	#16839538	C₄	1	1	yes	characteristic
16	#16839537	C₄	1	1	yes	characteristic
17	C₁₆×Dic₆₁	C₂²	1	1	yes	characteristic
18	C₂×C₃₂×C₆₁	C₂²	1	1	yes	characteristic
19	C₂×C₃₂×C₆₁	C₂²	1	1	yes	characteristic
20	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
21	C₂×#11758189	C₈	1	1	yes	characteristic
22	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
23	#14519734	C₈	1	1	yes	characteristic
24	#14519716	C₈	1	4	yes	—
25	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
26	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
27	C₁₆×D₆₁	C₂×C₄	1	2	yes	—
28	C₁₆×D₆₁	C₂×C₄	1	2	yes	—
29	#14519688	C₈	1	1	yes	characteristic
30	#14519687	C₈	1	1	yes	characteristic
31	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
32	#14519683	C₂×C₄	1	2	yes	—
33	#14519683	C₂×C₄	1	2	yes	—
34	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
36	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
37	C₃₂×C₆₁	C₂×C₄	1	2	yes	—
38	C₃₂×C₆₁	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₆₁	C₂×C₈	1	1	yes	characteristic
40	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
41	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
42	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
43	#11758198	C₁₆	1	2	yes	—
44	#11758196	C₁₆	1	2	yes	—
45	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
46	C₂×#92418	C₂×C₈	1	1	yes	characteristic
47	C₂×#92418	C₂×C₈	1	1	yes	characteristic
48	#11758189	C₁₆	1	4	yes	—
49	C₈×D₆₁	C₄²	1	2	yes	—
50	C₈×D₆₁	C₂×C₈	1	2	yes	—
51	C₁₆×C₆₁	C₂×C₈	1	2	yes	—
52	C₁₆×C₆₁	C₂²×C₄	1	1	yes	characteristic
53	C₁₆×C₆₁	C₂²×C₄	1	1	yes	characteristic
54	#11758185	C₄²	1	2	yes	—
55	#11758185	C₂×C₈	1	2	yes	—
56	C₂²×D₆₁	C₂×C₁₆	1	1	yes	characteristic
57	C₂×C₄×C₆₁	C₂×C₁₆	1	2	yes	—
58	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
59	C₂×Dic₆₁	C₂×C₁₆	1	2	yes	—
60	C₂×Dic₆₁	C₂×C₁₆	1	1	yes	characteristic
61	C₄×D₆₁	C₄×C₈	1	2	yes	—
62	C₄×D₆₁	C₂×C₁₆	1	2	yes	—
63	C₈×C₆₁	C₂×C₁₆	1	2	yes	—
64	C₈×C₆₁	C₂×C₄²	1	1	yes	characteristic
65	C₈×C₆₁	C₂²×C₈	1	1	yes	characteristic
66	#92418	C₄×C₈	1	2	yes	—
67	#92418	C₂×C₁₆	1	2	yes	—
68	#13549	—	61	1	no	maximal, Sylow
69	C₂²×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
70	C₂×D₆₁	—	2	2	no	—
71	C₂×D₆₁	C₄×C₁₆	1	2	yes	—
72	C₄×C₆₁	—	2	2	no	—
73	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
74	C₄×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
75	Dic₆₁	—	2	2	no	—
76	Dic₆₁	C₄×C₁₆	1	2	yes	—
77	C₂×M₆₄	—	61	1	no	—
78	C₂²×C₃₂	—	61	1	no	—
79	#2108	—	61	1	no	—
80	#1397	—	61	1	no	—
81	#1397	—	61	1	no	—
82	C₄×C₃₂	—	61	1	no	—
83	C₄×C₃₂	D₆₁	1	1	yes	characteristic, complemented
84	C₂×C₆₁	C₂×C₄×C₁₆	1	1	yes	characteristic
85	C₂×C₆₁	#2258	1	2	yes	—
86	D₆₁	—	2	2	no	—
87	C₂×M₃₂	—	61	1	no	—
88	C₂²×C₁₆	—	61	1	no	—
89	#405	—	61	1	no	—
90	M₆₄	—	61	4	no	—
91	C₂×C₃₂	—	61	1	no	—
92	C₂×C₃₂	—	61	1	no	—
93	C₂×C₃₂	—	61	4	no	—
94	C₂×C₃₂	C₂×D₆₁	1	1	yes	characteristic
95	C₂×C₃₂	C₂×D₆₁	1	1	yes	characteristic
96	#346	—	61	1	no	—
97	#346	—	61	1	no	—
98	C₄×C₁₆	—	61	1	no	—
99	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
100	C₆₁	#13549	1	1	yes	characteristic, complemented, Sylow
101	C₂×M₁₆	—	61	1	no	—
102	C₂²×C₈	—	61	1	no	—
103	#117	—	61	1	no	—
104	M₃₂	—	61	4	no	—
105	C₂×C₁₆	—	61	1	no	—
106	C₂×C₁₆	—	61	1	no	—
107	C₂×C₁₆	—	61	2	no	—
108	C₂×C₁₆	—	61	2	no	—
109	C₂×C₁₆	C₄×D₆₁	1	1	yes	characteristic
110	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
111	#97	—	61	1	no	—
112	#97	—	61	1	no	—
113	C₄×C₈	—	61	1	no	—
114	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
115	C₃₂	—	61	2	no	—
116	C₃₂	—	61	2	no	—
117	C₃₂	C₄×D₆₁	1	2	yes	—
118	C₃₂	C₄×D₆₁	1	2	yes	—
119	C₂²×C₄	—	61	1	no	—
120	M₁₆	—	61	4	no	—
121	C₂×C₈	—	61	1	no	—
122	C₂×C₈	—	61	1	no	—
123	C₂×C₈	—	61	2	no	—
124	C₂×C₈	—	61	2	no	—
125	C₂×C₈	C₈×D₆₁	1	1	yes	characteristic
126	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
127	C₄⋊C₄	—	61	2	no	—
128	C₂²⋊C₄	—	61	2	no	—
129	C₄²	—	61	1	no	—
130	C₄²	C₈×D₆₁	1	1	yes	characteristic
131	C₁₆	—	61	2	no	—
132	C₁₆	—	61	2	no	—
133	C₁₆	C₈×D₆₁	1	2	yes	—
134	C₁₆	C₂×C₄×D₆₁	1	1	yes	characteristic
135	C₁₆	C₂×C₄×D₆₁	1	1	yes	characteristic
136	C₂³	—	61	1	no	—
137	C₂×C₄	—	61	1	no	—
138	C₂×C₄	—	61	2	no	—
139	C₂×C₄	—	61	2	no	—
140	C₂×C₄	—	61	2	no	—
141	C₂×C₄	C₁₆×D₆₁	1	2	yes	—
142	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
143	C₈	—	61	2	no	—
144	C₈	—	61	2	no	—
145	C₈	C₁₆×D₆₁	1	2	yes	—
146	C₈	C₄²×D₆₁	1	1	yes	characteristic
147	C₈	C₂×C₈×D₆₁	1	1	yes	characteristic
148	C₂²	—	61	2	no	—
149	C₂²	—	122	2	no	—
150	C₂²	C₂×C₁₆×D₆₁	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	61	2	no	—
153	C₄	—	122	2	no	—
154	C₄	C₄×C₈×D₆₁	1	1	yes	characteristic
155	C₄	C₂×C₁₆×D₆₁	1	1	yes	characteristic
156	C₂	—	122	2	no	—
157	C₂	C₄×C₁₆×D₆₁	1	1	yes	characteristic
158	C₂	#21448112	1	2	yes	—

	subgroup	G/H
center	C₂×C₃₂	C₂×D₆₁
commutator	C₂×C₆₁	C₂×C₄×C₁₆
Frattini	C₂×C₁₆	C₂²×D₆₁
Fitting	C₄×C₃₂×C₆₁	C₂
socle	C₂²×C₆₁	C₂²×C₁₆
prime radical	C₂²×D₆₁	C₂×C₁₆
cosocle	C₂×C₁₆×C₆₁	C₂³
hypercenter	C₄×C₃₂	D₆₁
hypocenter	C₆₁	#13549
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₃₂ ◁ C₄×C₃₂

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
4. #21443953	C₂	semidirect
5. #21443952	C₂	semidirect
6. C₃₂×Dic₆₁	C₂	semidirect
7. C₄×C₃₂×C₆₁	C₂	semidirect
11. #16839595	C₄	semidirect
14. C₃₂×D₆₁	C₄	semidirect
83. C₄×C₃₂	D₆₁	semidirect
100. C₆₁	#13549	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#16839595	2
C₂×C₃₂×D₆₁	2
#21447414	2
#21443953	2
#21443952	2
C₃₂×Dic₆₁	2
C₄×C₃₂×C₆₁	2
#13549	61